Use the Extended Euclidean algorithm.
Show that z gcd(x,y) is a common divisor of zx and zy that can be written as a linear combination of zx and zy.
Try to make the divisions.
Imitate the proof of the theorem stating that there are infinitely many primes.
What can you say about the remainder r of the division of c by lcm(a,b)? Notice that r is divisible by both a and b.
Notice that n2 - 1 = (n - 1)(n + 1).
Find a factorization of numbers of the form n2 - 1.
Use the extended Euclidean algorithm.
Suppose that the cube root of 17 is a rational number a/b, with a and b integers. Then 17b3 = a3. Deduce a contradiction by looking at ord17 of both left- and right-hand side.
Use that a2 - 1 = (a - 1)(a + 1).
Use division with remainder or the formula a3 - 1 = (a - 1)(a2 + a + 1).
Write m = 2n + 1 and square.
If a divides b, then we can write b as qa for some integer q. Use this.
Note that 18 and 25 are relatively prime and use the extended Euclidean algorithm.
Find a particular solution, and then solve the homogeneous equation.
Rewrite the equation as p(q - 4) = 7q and investigate the factors on both sides.